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arXiv:2307.03062

Quantum criticality on a compressible lattice

Saheli Sarkar, Lars Franke, Nikolas Grivas, Markus Garst

Published Jul 6, 2023• 0 citations
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Abstract

The stability of a quantum critical point in the $O(N)$ universality class with respect to an elastic coupling, that preserves $O(N)$ symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension $d=3-ε$. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for $1\leq N \leq 4$, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length $ξ$. For $N > 4$, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached the sound velocity is found to vanish only asymptotically as $ξ\to \infty$ such that the crystal remains microscopically stable for any finite value of $ξ$. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in $d=4-ε$, where the crystal always remains microscopically stable for finite $ξ$.

DOI
10.48550/arxiv.2307.03062

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Quantum criticality on a compressible lattice

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The stability of a quantum critical point in the $O(N)$ universality class with respect to an elastic coupling, which preserves $O(N)$ symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension $d=3\ensuremath{-}\ensuremath{\epsilon}$. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for $1\ensuremath{\le}N\ensuremath{\le}4$, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length $\ensuremath{\xi}$. For $N>4$, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached, the sound velocity is found to vanish only asymptotically as $\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}$, such that the crystal remains microscopically stable for any finite value of $\ensuremath{\xi}$. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in $d=4\ensuremath{-}\ensuremath{\epsilon}$, where the crystal always remains microscopically stable for finite $\ensuremath{\xi}$.

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The stability of a quantum critical point in the O(N)O(N)O(N) universality class with respect to an elastic coupling, that preserves O(N)O(N)O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension d=3−εd=3-εd=3−ε. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for 1≤N≤41\leq N \leq 41≤N≤4, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length ξξξ. For N>4N > 4N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached the sound velocity is found to vanish only asymptotically as ξ→∞ξ\to \inftyξ→∞ such that the crystal remains microscopically stable for any finite value of ξξξ. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in d=4−εd=4-εd=4−ε, where the crystal always remains microscopically stable for finite ξξξ.
The stability of a quantum critical point in the O(N)O(N)O(N) universality class with respect to an elastic coupling, which preserves O(N)O(N)O(N) symmetry, is investigated for isotropic elasticity in the framework of the renormalization group (RG) close to the upper critical dimension $d=3\ensuremath{-}\ensuremath{\epsilon}$. With respect to the Wilson-Fisher fixed point, we find that the elastic coupling is relevant in the RG sense for $1\ensuremath{\le}N\ensuremath{\le}4$, and the crystal becomes microscopically unstable, i.e., a sound velocity vanishes at a finite value of the correlation length $\ensuremath{\xi}$. For N>4N>4N>4, an additional fixed point emerges that is located at a finite value of the dimensionless elastic coupling. This fixed point is repulsive and separates the flow to weak and strong elastic coupling. As the fixed point is approached, the sound velocity is found to vanish only asymptotically as $\ensuremath{\xi}\ensuremath{\rightarrow}\ensuremath{\infty}$, such that the crystal remains microscopically stable for any finite value of $\ensuremath{\xi}$. The fixed point structure we find for the quantum problem is distinct from the classical counterpart in $d=4\ensuremath{-}\ensuremath{\epsilon}$, where the crystal always remains microscopically stable for finite $\ensuremath{\xi}$.
The Holstein model with dispersionless Einstein phonons is one of the simplest models describing electron-phonon interactions in condensed matter. A naive extrapolation of perturbation theory in powers of the relevant dimensionless electron-phonon coupling ${\ensuremath{\lambda}}_{0}$ suggests that at zero temperature the model exhibits a Pomeranchuk instability characterized by a divergent uniform compressibility at a critical value of ${\ensuremath{\lambda}}_{0}$ of order unity. In this work, we re-examine this problem using modern functional renormalization group (RG) methods. For dimensions d>3d>3d>3 we find that the RG flow of the Holstein model indeed exhibits a tricritical fixed point associated with a Pomeranchuk instability. This non-Gaussian fixed point is ultraviolet stable and is closely related to the well-known ultraviolet stable fixed point of ${\ensuremath{\phi}}^{3}$-theory above six dimensions. To realize the Pomeranchuk critical point in the Holstein model at fixed density both the electron-phonon coupling ${\ensuremath{\lambda}}_{0}$ and the adiabatic ratio ${\ensuremath{\omega}}_{0}/{\ensuremath{\epsilon}}_{F}$ have to be fine-tuned to assume critical values of order unity, where ${\ensuremath{\omega}}_{0}$ is the phonon frequency and ${\ensuremath{\epsilon}}_{F}$ is the Fermi energy. However, for dimensions $d\ensuremath{\le}3$ we find that the RG flow of the Holstein model does not have any critical fixed points. This rules out a quantum critical point associated with a Pomeranchuk instability in $d\ensuremath{\le}3$.
Abstract Chapter 7 is devoted to a discussion of the renormalization group (RG) flow when the effective field theory that describes universal properties of critical phenomena depends on several coupling constants. The universal properties of a large class of macroscopic phase transitions with short range interactions can be described by statistical field theories involving scalar fields with quartic interactions. The simplest critical systems have an O(N) orthogonal symmetry and, therefore, the corresponding field theory has only one quartic interaction. However, in more general physical systems, the flow of quartic interactions is more complicated. This chapter examines these systems from the RG viewpoint. RG beta functions are shown to generate a gradient flow. Some examples illustrate the notion of emergent symmetry. The local stability of fixed points is related to the value of the scaling field dimension.
We study the fate of quantum solids with a vanishing velocity of transverse phonons. Such ``floppy'' solids exhibit a cubic elastic nonlinearity that is relevant in spatial dimensions d<4d<4d<4 and drives the system to a strongly-coupled Lifshitz quantum critical point in d=3d=3d=3, that we analyze in an ε=4−dε= 4-dε=4−d expansion. The associated phase transition is generically first-order, preempting the system's full access to the strongly-coupled fixed point. Nevertheless, the critical data of this quantum critical point is accessible within a Ginzburg region in the vicinity of the transition. This cubic nonlinearity vanishes exactly in d=2d=2d=2, where the leading interactions are instead quartic and are marginal, leading to weak logarithmic singularities. We discuss possible realizations of this ``crystal Lifshitz criticality'' in quantum materials.
Using the local potential approximation of the exact renormalization group (RG) equation, we show various domains of values of the parameters of the O(1) -symmetric scalar Hamiltonian.In three dimensions, in addition to the usual critical surface S c (attraction domain of the Wilson-Fisher fixed point), we explicitly show the existence of a first-order phase transition domain S f separated from S c by the tricritical surface S t (attraction domain of the Gaussian fixed point).S f and S c are two distinct domains of repulsion for the Gaussian fixed point, but S f is not the basin of attraction of a fixed point.S f is characterized by an endless renormalized trajectory lying entirely in the domain of negative values of the ϕ 4 -coupling.This renormalized trajectory also exists in four dimensions making the Gaussian fixed point ultra-violet stable (and the ϕ 4 4 renormalized field theory asymptotically free but with a wrong sign of the perfect action).We also show that a very retarded classical-to-Ising crossover may exist in three dimensions (in fact below four dimensions).This could be an explanation of the unexpected classical critical behaviour observed in some ionic systems.
Abstract In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field approximation, which is also a quasi-Gaussian approximation. The mean-field approximation predicts a set of universal properties, properties independent of the detailed structure of the microscopic Hamiltonian, the dimension of space, and, to a large extent, of the symmetry of systems. However, the leading corrections to the mean-field approximation, in dimensions smaller than or equal to four, diverge at the critical temperature, and the universal predictions of the mean-field approximation cannot be correct. Such a problem originates from the non-decoupling of scales and leads to the question of possible universality. In Chapter 9, the question has been answered in four dimensions using renormalization theory, and related renormalization group (RG) equations. Moreover, below four dimensions, in an expansion around the mean-field, the most singular terms near criticality can be also formally recovered from a continuum, low-mass φ4 field theory. More generally, following Wilson, to understand universality beyond the mean-field approximation, it is necessary to build a general renormalization group in the form of flow equations for effective Hamiltonians and to find fixed points of the flow equations. Near four dimensions, the flow equations can be approximated by the renormalization group of quantum field theory (QFT), and the fixed points and critical behaviours derived within the framework of the Wilson-Fisher ϵ expansion.